NAVAL POSTGRADUATE SCHOOL JMonterey, California

DTIC

IHESISPREDICTION OF ATTITUDE STABILITY OF

ASYMMETRIC DUAL-SPIN STABILIZED SPACECRAFTUSING IMPROVED LIQUID SLOSH MODEL

by

Michael J. Szostak

June 1991

Thesis Advisor: Prof. Brij Agrawal

Approved for public release; distribution unlimited

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11 Title (Include Security Class 'ication) Prediction of Attitude Stability of Asymmetric Dual-Spin Stabilized SpacecraftUsing Improved Liquid Slosh Model12 Personal Author(s) Michael J. Szostak13a Type of Report 13b Time Covered 14 Dae of Report (year, month,day) I 15 Page CountMaster's Thesis jFrom To IJune 1991 15216 Supplementary Notation The views expressed in this thesis are those of the author and do not reflect the officialpolicy or position of the De )artment of Defense or the U.S. Government.17 Cosati Codes 18 Subject Terms (continue on reverse if necessary and identify by block number)

Field Group Subgroup Dual-Spin Spacecraft; Liquid Fuel Model; Rotor and Platform Asymmetry

19 Abstract (continue on reverse if necessary and identify by block numberThe "'rigid slug" method for modelling sloshing liquid fuel aboard dual-spin stabilized spacecraft has been

shown to be inadequate by recent flight data. This "rigid slug" model and a uniform gravity model put forth byAbramson is examined in detail. The Abramson model is incorporated into a computer simulation writtenspecifically to predict spacecraft attitude. An analysis is performed with both the modified and unmodifiedversions of this simulation to determine the boundaries of stability for rotor and platform asymmetries. Theresults show that the improved model is better able to predict spacecraft attitude stability.

20 Distribution/Availability of Abstract 21 Abstract Security Classification] unclassified/unlimited []ame as rport Drc usen Unclassified

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Approved for public release; distribution is unlimited.

Prediction of Attitude Stabilityof Asymmetric Dual-Spin Stabilized Spacecraft

Using Improved Liquid Slosh Model

by

Michael J. SzostakLieutenant Commander, United States NavyB.S., United States Naval Academy, 1979

Submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN ASTRONAUTICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

June 1991

Author:

Approved By: (ABrij N. Agrawal, Thesis Advisor

I. Michael Ross, Sec d Reader

E. R. Wood, Chairman, Depr ent of Aeronautics andAstronautics

ii

ABSTRACT

The "rigid slug" method for modelling sloshing liquid fuel aboard dual-spin

stabilized spacecraft has been shown to be inadequate by recent flight data. This

"rigid slug" model and a uniform gravity model put forth by Abramson is

examined in detail. The Abramson model is incorporated into a computer

simulation written specifically to predict spacecraft attitude. An analysis is

performed with both the modified and unmodified versions of this simulation to

determine the boundaries of stability for rotor and platform asymmetries. The

results show that the improved model is better able to predict spacecraft attitude.

Accession For

NTIS -,A&I

DTTC T4B

Av . J : .21ty Ccjios

';i il

iiiJ _/ _

TABLE OF CONTENTS

1. IN TR O D U CTIO N ................................................................................ 1

A . BA CK G RO U N D ........................................................................... 1

B. O BJECTIVE ............................................................................ 2

C. LITERATURE REVIEW .......................................................... 3

D. ORGANIZATION OF STUDY .................................................. 5

II. BACKGROUND .............................................................................. 7

A. DUAL-SPIN SATELLITES ........................................................ 7

B. ENERGY SINK DERIVATION .................................................. 8

C. SIMULATION DESCRIPTION ............................................... 13

1. Input .............................................................................. .. 15

2. O utput ............................................................................ 15

D. SATELLITE CONFIGURATION ............................................. 16

III. LIQUID SLOSH MODELS .............................................................. 19

A. RIGID SLUG MODEL ............................................................. 19

B. UNIFORM GRAVITY MODEL ................................................ 20

IV. PROCEDURE ............................................................................... 25

A. SYSTEMS PARAMETERS FOR ASYMMETRY ANALYSIS ......... 25

B. SIMULATION PROCEDURES FOR ASYMMETRY ANALYSIS... 28

C. SIMULATION PROCEDURES FOR MODEL ENHANCEMENT... ? 1

V. RESULTS AND ANALYSIS ........................................................... 33

A. SIMULATION RESULTS FOR ROTOR ASYMMETRY ........... 33

B. SIMULATION RESULTS FOR PLATFORM ASYMMETRY ....... 35

C. ANALYSIS ........................................................................... 36

VI. SUMMARY AND CONCLUSIONS ................................................. 40

A. SUMMARY ........................................................................... 40

B. CONCLUSIONS ..................................................................... 40

iv

LIST OF REFERENCES......................................................... 42

INITIAL DISTRIBUTION LIST ................................................ 43

LIST OF TABLES

TABLE 1. Mass and Inertia Properties of the Rotor and Platform (Dry) ..... 18

TABLE 2. Liquid Fuel Characteristics ................................................. 18

TABLE 3. Wet Spacecraft Parameters for Fuel Modelled as Spherical

Pendulum s ......................................................................... 26

TABLE 4. Wet Spacecraft Parameters for Fuel Modelled as Spherical

Pendulums (Modified Simulation) ........................................ 32

TABLE 5. Frequencies and Pendulum Arm Lengths of Models ............... 38

TABLE 6. Recommended and Actual Pendulum Arm Lengths ................. 38

vi

LIST OF FIGURES

Figure 1. Idealized Dual-Spin Stabilized Spacecraft ................................. 8

Figure 2. Dual Spin Spacecraft with Liquid in Spherical Tanks .............. 14

Figure 3. Fuel Tank Arrangement for INTELSAT VI ........................... 17

Figure 4. Liquid Fuel Modelled as a Spherical Pendulum ...................... 20

Figure 5. Abramson's Analytical Model ............................................. 21

Figure 6. Abramson's Curves for Analytical Model [Ref. 8] .................. 22

Figure 7. Normalized First Mode Frequency Comparison ..................... 23

Figure 8. Normalized First Mode Mass Comparison ............................. 24

Figure 9. Flowchart for Rotor Asymmetry Analysis Procedure ............. 30

Figure 10. Stability Cutoff Points for Inertia Ratios 1.03, 1.05, 1.07 and

1.1 .................................................................................... 33

Figure 11. Stability Cutoff Points for Inertia Ratios 1.03, 1.05, 1.07 and

I . 1. (M odified Simulation) .................................................. 34

Figure 12. Minimum Inertia Ratio for Stability (Rotor Asymmetry) ........... 35

Figure 13. Minimum Inertia Ratio for Stability (Platform Asymmetry) ....... 36

Vii

viii

I. INTRODUCTION

A. BACKGROUND

Spin stabilized spacecraft have been in existence almost from the very start

of man's artificial satellite program. This type of attitude control for satellites

uses an inherently simple principle, that is, to provide gyroscopic stiffness for

inertial pointing by rotation of the whole or part of the satellite. "Dual spin"

satellites are a subset of spin stabilized spacecraft and have two bodies, a rotor

and a platform, that spin about a common axis. The rotor spins at a high angular

rate to provide inertial pointing while the platform generally spins at a rate of

one revolution per orbit period to be earth pointing.

Under the current trend of using liquid fuelled apogee motors, liquid

propellant constitutes almost two thirds of rotor mass in transfer orbit. With

fuel loads now comprising such a high fraction of spacecraft mass, it is critical

that fuel slosh effects be better understood. Just as flexible elements on the

spacecraft can interact stably or unstably, so can sloshing fuel interact stably or

unstably.

Recent flight data from the first two LEASAT communications satellites

have raised questions about the current modelling of sloshing liquid fuel. Both

satellites experienced instability during transfer orbit that was not predicted by

the computer model used. The simulation that was used models liquid propellant

as spherical pendulums attached to the centers of the fuel tanks. The principals

behind this model need to be examined.

B. OBJECTIVE

The primary objective of this study is to improve the modelling of liquid

propellant aboard dual spin stabilized spacecraft. This will be accomplished by

using an analytical model of sloshing liquid put forth by Abramson [Ref. 11].

Abramson's model, called the uniform gravity model, treats liquid as a pendulum

whose parameters such as mass of the pendulum, length of the pendulum's arm

and frequency of oscillation are determined by fill fraction of the tank. This

model is different than most in that only a portion of the liquid in the tank is

modeled as a pendulum and that the rest of the fuel as an addition to the dry

moment of inertia of the rotor. Abramson's model will be incorporated into a

computer simulation authored by Chung [Ref. 21. Chung's computer program

was originally intended to model the attitude of INTELSAT VI, but it can be

adapted to model the attitude of almost any dual-spin spacecraft with large liquid

mass. The liquid model that Chung used for his simulation is nearly identical to

the one used for the LEASAT flights. This model, named the "rigid slug"

model, depicts liquid fuel as spherical pendulums with two degrees of freedc rn

and assumes that 100 percent of the liquid propellant is in motion. The goal of

this study is to incorporate the more accurate Abramson model into Chung's

simulation.

The second objective of this study is to explore the boundary between

stability and instability of dual-spin stabilized spacecraft with asymmetric rotors

and platforms. Specifically, it is highly desirable to know whether liquid slosh is

stabilizing or not for various degrees of rotor and platform asymmetry, inertia

ratios and fuel loads. Knowing this point of stability/instability, engineers will

know the upper limit in asymmetry of the platforms or the rotors, and satellite

controllers can more accurately predict attitude control requirements.

2

To perform the rotor asymmetry analysis, Chung's computer simulation with

the "rigid slug" model of liquid fuel will be used along with a version of the

simulation modified with Abramson's model. Chung's simulation, using

equations of motion to model spacecraft attitude as a function of time, can

predict stability. Starting with a stable spacecraft, rotor asymmetry will be

increased until "astability occurs. This point will be called the stability cutoff

point. For the study, inertia ratio is the ratio of the spin axis moment of inertia

to the average of the transverse moments of inertia. Instability is defined where

the spacecraft nutation angle does not decrease passively. The entire analysis will

be performed with both the modified and unmodified versions of Chung's

simulation.

To perform the platform asymmetry analysis, only Chung's modified

simulation will be used. The objective of this portion of the study will be the

determination .of the stability boundary limits of an asymmetric platform and an

axisymmetric rotor.

C. LITERATURE REVIEW

In 1963, W.T. Thomson [Ref. 31 presented a paper on the motion of a

rotating asymmetric body with internal dissipation. Noting that this moment free

motion resulted in elliptical functions, Thomson presented the results in a form

very similar to the symmetric case. The solution stated the nutation angle rate is

a function of kinetic energy rate, angular momentum, and the moments of

irertia.

Peter Likins [Ref. 4] in 1967, was the first to extend initial spin stability

analysis to non axisymmetric vehicles. All previous work to that point involved

systems with symmetrical platforms and rotors. By Routhian analysis though,

Likins showed that a system with an asymmetric despun platform could be made

3

asymptotically stable. Energy losses in the rotor were found to cause instability

if rotor spin inertia was less than the average total transverse inertia. This is the

famous energy sink stability criteria, that is, the ratio of the axial moment of

inertia of the rotor to the average of the transverse moments of inertia of the

spacecraft must be greater than one.

T.M. Spencer [Ref. 5] in 1973 challenged Likins on several points and

questioned his assumptions concerning the energy sink derivation. Specifically,

Spencer believed that using the algebraic energy sink equation is an

oversimplification for asymmetric satellites. Spencer stated that the kinetic

energy dissipation rate in the rotor is a time varying factor, and that variations

must be determined before averaging can be used to get a stability result.

Spencer concluded by proposing that the geometric mean transverse moment of

inertia is a crucial stability parameter, rather than the algebraic mean as

forwarded by Likins.

Chung, in 1985, stated that previous works applying the energy sink equation

to dual spin spacecraft made certain assumptions that are not applicable for

spacecraft with a high liquid fraction. Specifically, the general equation for

angular momentum does not take into account accurately the effects of liquid

slosh. He added that the equation for kinetic energy does not always decrease

and concluded with a warning that the energy sink results be used with caution

for spacecraft with large liquid fuel loads and high inertia asymmetries. Chung

authored a computer simulation for spacecraft attitude based on the "rigid slug"

model at this time.

In 1987, Slafer [Ref. 6] showed that the "rigid slug" model like the one used

by Chung to model dual-spin spacecraft attitude was not accurate for spacecraft

with high liquid fuel loads. Slafer analyzed the interaction between the control

system and the large amount of liquid propellant aboard the first two LEASAT

4

synchronous orbit communications satellites. Slafer's post flight analysis

included several simulations and compared them with actual flight data. The

"rigid slug" model that had been previously used at Hughes for static stability

analysis, failed to predict the observed flight instability. Abramson's model

however, correctly predicted the observed flight instability.

Myers [Ref. 7], in 1990, used Chung's simulation to show that an inertia ratio

greater than one was not alone criteria for passive stability, but that the inertia

ratio had to be greater than one by a certain amount. This stability point was

dependent on fuel load and platform asymmetry. Myers also showed that

increasing platform asymmetry increased stability.

A simple algebraic expression like the energy sink equation has shown itself

useful for first order stability approximations. This is especially true if a dual

spin satellite can be modelled as a rigid, rotating body. To use this simple

equation as a definitive stability parameter for high liquid fuel spacecraft though,

is circumspect. The energy sink equation therefore,will only be used as an initial

starting point in the rotor and platform asymmetry analysis portion of this study.

Chung's simulation program will be used to further define the boundaries of

passive stability for rotor and platform asymmetry as a function fuel load and

inertia ratio. Slafer's results bring into question Chung's "rigid slug" method of

modelling liquid and therefore several enhancements to Chung's simulation will

be proposed based on the uniform gravity model.

D. ORGANIZATION OF STUDY

Chapter II of this study will further define dual spin satellites and also

provide a derivation of the energy sink equation. A description of Chung's

computer simulation as well as the required inputs and output for the program

are provided in Section C of this chapter. The various satellite configurations

5

used in the study are described in Section D. Abramson's model and the

proposed changes to Chung's simulation are provided in Section E.

Chapter III presents a description of both the "rigid slug" and the uniform

gravity model. Detailed graphics, and equations and curves for the computation

of parameters are included. Lastly, Slafer's results comparing the two models

with the actual LEASAT flight data are presented.

Chapter IV contains the procedures to generate the input data and the

procedures to run the main simulation program. The spreadsheet written to

generate platform and rotor moments of inertia for a given inertia ratio and fuel

load is described in Section A. Also described in Section A is a program written

by Chung to convert INTELSAT VI parameters to a format readable by the

main simulation program. Section B describes the procedures used for the rotor

and platform asymmetry analysis and Section C describes how Abramson's

model is incorporated into Chung's simulation.

Chapter V presents the results of the rotor asymmetry analysis for both the

modified and unmodified computer simulations. Plots are provided for each of

the models to show the location of the stability cutoff points For the platform

asymmetry analysis, the minimum inertia ratio for zero, five and ten percent

platform asymmetry is determined. An analysis is performed to explain the

curves.

Chapter VI presents the summary of the findings and the conclusions.

6

II. BACKGROUND

This chapter will clarify terms and definitions of the analysis of rotor and

platform asymmetry of dual-spin stabilized spacecraft. A derivation of the

energy sink equation and a description of the simulation program authored by

Chung will be provided. The method used to derive the various satellite

configurations, as well as several improvements to Chung's simulation will also

be discussed.

A. DUAL-SPIN SATELLITES

"Dual-spin" spacecraft are comprised of two primary bodies capable of

unlimited relative rotation about a common axis. Typically, the spun section is

called the rotor while the despun body the platform. A motor is usually mounted

on the platform which is used to drive the rotor so that it maintains a constant

rotation rate relative to the platform. The rotation speed of the rotor is

dependent on the moments of inertia and required accuracy in orientation of the

spacecraft. The platform can be either completely despun or rotate at a rate as

required by the payload. The utility of such a combination of bodies allows the

simplicity and reliability of spin stabilization to be used with the unidirectional

pointing of components like sensors and antennas. Dual-spin satellites vary in

size from the 70 kg INTELSAT I to the 12,000 kg LEASAT.

As the size of dual-spin satellites has increased over the years, so has the

amount of fuel these satellites are required to carry. In addition, extended

lifetimes also have necessitated larger fuel loads. Spacecraft fuel is normally

located in spherical tanks on the rotor that are extended out from the axis of

7

rotation. The number and size of the tanks is a function of the mission and

lifetime of the satellite. An idealized dual-spin system is depicted in Figure I.

V

I

IRotor

I

IIIIz|

Platform

Figure 1. Idealized Dual-Spin Stabilized Spacecraft

B. ENERGY SINK DERIVATION

The derivation of the stability condition is provided using Likins' model of

an idealized system depicted in Figure 1. The figure depicts a platform P of

arbitrary inertia properties and an axisymmetric rotor R. The center of mass of

both bodies lies along the axis Y, the axis of rotation. As a starting point the

equation of motion governing the entire system S is

M= h (1)

8

where M is the applied moment about the system mass center 0, and h the

angular momentum about 0. The dot over the vector h depicts time rate of

change in inertial space. The resulting linearized Euler equations of motion are

MI 116)1 + (13 - 12) Op(0 2 + Ir0r(02 = 0 (2)

M2= 126)2 + (II - 13) 0)p0 1 - Ir0)r031 = 0 (3)

M3 = 13)p + Ir 6)r + 120)102 - I10102 = 0 (4)

Mr = Ir ()p + 60r) (5)

where MI, M2 and M3 are the resulting moments about each of the primary axes

and Mr the moment applied about the rotor by the platform. I1 and 12 are the

transverse moments of inertia of the system while 13 is the system's axial moment

of inertia. Ir is the axial moment of inertia of the rotor. The transverse angular

rates of the system are col and 0)2 while wop is the rotation rate of the platform

and 0Wr the relative rotation rate of the rotor with respect to the platform. The

four unknowns, 0)1, (02, (op and 0 r can now be solved for using the four

equations.

Equations (2) and (3) can be further simplified under the assumption that the

system consists of a rigid asymmetric body with a rigid axisymmetric rotor

attached, and no external moments where

X1 = ((13 -1 2) 0)p + IrOr)/11 (6)

).2 = ((13 -11) 0p + Ir)/A2 (7)

Eqs. (2) and (3) become

61 +X -1(02 =0 (8)

6)2 - -20 1 =0 (9)

Changes in rotation rates of the two bodies about their common axis are

ignored assuming small motor and bearing friction torques. Assuming the initial

conditions

0-2(0) = 00 0)1(0) = 0 (10)

9

the solutions to (8) and (9) become

(01 = .COO " sin( -5 / -X2 *t0(1

()2 = (o0 * cos(VXI.X 2 *t) (12)

The resulting necessary condition for stability is

X * X2 > 0 (13)

The kinetic energy for this dual spin system is

T = 1/211o)12 + 1/2120)22 + 1/2130)p 2 + 1/2IrC0r2 + Ir"Qp (14)

and the square of the system angular momentum about 0 is

h2 = I12(012 + 122)22 + (l3top + Ir(or)2 (15)

The angular momentum magnitude of the nominal motion is

ho = 13(Op + Itor (16)

The time derivative of Eqs. (14) and (15) yields the following equations. Note

that the kinetic energy rate is negative because of an assumed energy dissipating

device on the system and that the angular momentum is constant

d/dt (T)= 11(0I) 1 + 12o)2(02+ I3top0p + lrortir+ Irtor(p+ lrop(r<O (17)

1/2 d/dt (h) = l2(old, + 122oj)262 + (13t)p + lr(Ir)(I36)p + lr(br) = 0 (18)

The solutions to the Euler equations (Eqs. 11 and 12) are no longer valid

under the just mentions conditions. However, if the effects of the energy

dissipating device are felt over time, the form of Eqs. (11) and (12) is correct

with oo taking the form of oo = coo(t), a slowly varying function of time.

Averaging over the period ' = 21/2X2(Xl/X 2)o-5 and assuming slow time

variations of o, the approximation for the kinetic energy becomes

T= '/211to0OO(X A 2) + l/212(o0 + 13(p(i0p

+ lro0r(6Or + Iraw p+ 1to0p(Or (19)

10

and the moment equation0 = 1/211200*0 (X.I/'k2)+ 122(,006)0

+ (13(Op + lrOr)(136)p + Irir) (20)

To obtain the kinetic energy rate in terms of the platform and rotor angular

rates, it is first necessary to solve Eq. (20) for o06oo. This results in the

following

0060 = -2X2 (I3C0p+Ir&)r)(1 36)p+lrObr)/(1 12 1 +122X2) (21)

Substituting this into Eq. (19) yieldsT= -I pd)p(-(Op+(I3(0p+IrOr)*(I 1 .+I2X2)/(112XI+I22X2))

-Ir(6 r+d)p)*(-((or+(I )+(I3(0p+Ir(Or)*( l X i+I2X2)/(1,2xl +I22).2)) (22)

where Ip is the axial moment of inertia of the platform. Recalling the definition

of the nominal angular momentum magnitude, the expression for T can be

further simplified using the following substitution

ko = ho*(I )X-1+12 X2)/(I 2)X1+122X2 ) (23)

which results in

T = -Ipd)P0'p)-Ir(6r4)pX o-(or+ Op)) (24)

Using the following definitions in order to simplify

= X-Wp and Xr= Xo-(Or+Wp) (25)

the approximation for kinetic energy rate becomes

T = -IpdpXp - Ir(1)r+(4p)kr (26)

The two terms on the right side of Eq. (26) can be written as

P = -IP pXp and Pr =- lr(6)r+6Op)Xr (27)

which are the first approximations for the average energy dissipation rates for

the platform and the rotor. These equations can be rewritten in order to be

substituted into Eq. (21).

Ip6p = -Pp/XP Irr4<p) = -PrAr (28)

11

The substitution yields

_opo+ Pr) (21X2) (29)xp +r (I12 1+122X2)

As a necessary and sufficient condition for stability

(0o60 < 0 (30)

and since XlX2 > 0 was shown to be necessary for stability, and ho > 0 by

convention, then Eqs. (29) and (30) together result in

P< + <0 (31)Xp Xr

Since both the energy dissipating rates of the platform and the rotor must be

negative, both Xp and X must be positive or if one is negative, the respective

energy dissipating rate must be large enough so that the condition of Eq. (31) is

satisfied.

There are several special cases to study, but for the purposes of this analysis

we will consider the case of a despun platform, for which the rotation speed of

the platform ,op will be zero. With this assumption Eqs. (6) and (7) simplify to

X1 = Ir(Or/11 (7)

X2 Ir(Or/12 (8)

Substitution into Eq. (20) will simplify the value for Xo and result in the

following equation20 ro(

( + 12) (20)Substituting this value into Eq. (23)

X=(I1 + 12) - (5

Factoring out an or results in2IA - 1) (25)

(11 + 12)

For this case recall that o)p is zero and assume that Pp also equals zero. Eq. (31)

then becomes

12

If P, < 0, that is assume damping is on the rotor(ie liquid fuel), then Xr must be

positive. Therefore Eq. (25) then becomesS(I+ 2) 1)>0 (25)

Dividing both sides by (or and adding one to both sides results in the very

familiar energy sink equation2r > 1 (32)

(I1 + 12)Likins' derivation cited above does not take into account asymmetric rotors,

but it can be shown that the form is the same as Eq. (32). The conclusion drawn

from this fact is that it does not matter in which body the damping mechanism is

located. In addition, Likins proved that a body could rotate about its axis of least

inertia and be asymptotically stable, but fell short of proving that the energy sink

equation had application for large liquid loads (because of the simplifying

equations made). Because of its widespread use in industry, the criteria put forth

in Eq. (32) will be used in this analysis to as an initial stability parameter.

C. SIMULATION DESCRIPTION

The simulation program authored by Chung can model the motion of almost

any dual-spin spacecraft. The actual modeling consists of determining certain

time dependent variables given initial conditions for these variables. The

simulation can accommodate fuel tanks on either the platform or the rotor. The

size, location and number of tanks is determined by the simulation user. A

schematic representation of the dual spin stabilized satellite used by Chung is

shown in Figure 2.

Using the model above, differential equations governing generalized coordinates

and generalized speeds that characterize all motions of the spacecraft are written.

The differential equations of motion are of the form.

xi = fj(xj,...,xT) where i = I,...n

13

where f1, f2,...fn are in general nonlinear functions of their arguments. These

functions are assumed to be continuous and to have continuous derivatives The

equations are solved numerically in a subroutine of the main simulation

program.

DUAL SPIN SPACECRAFT

Common Axis _ % i

Dry Spacecraft Liquid Fuel

i Rotor

on the rotor

S on the platform

Viscous Spherical Joints

Figure 2. Dual Spin Spacecraft with Liquid in Spherical Tanks

Kane's method is used to obtain the exact equations of motion for the model

spacecraft. This method involves the following three steps

1. The derivation of expressions for generalized inertia forces.

2. The derivation of expressions for generalized active forces.

3. Substitution into the equation

Fr + Fr* = 0 (r = 1, 2,...Nu)

14

where Fr and Fr* are the generalized active forces and the generalized inertia

forces and Nu is the total number of generalized speeds. In addition to the Nu

equations of motion, kinematical equations governing generalized coordinates are

also required.

The generalized active forces of the system are computed by summing all

active forces on each part of the system. The parts of the system that are

considered are the platform, rotor, and the spherical pendulum of the platform

and/or rotor. It is assumed that there are no external forces acting on any of the

spherical pendulums and that the resultant external forces on the platform and on

the rotor are zero.

The generalized inertia forces of the system are also computed by

considering contributions from the platform, rotor and the spherical pendulums

on the platform and/or rotor.

The kinematical equations of motion are the equations governing the

generalized speeds. These equations are primarily first order differential

equations governing the motion of the spacecraft.

1. Input

The required inputs into the simulation include mass and dry moments

of inertia of the platform and the rotor, as well as their center of mass position

from a reference point. Fuel tank data required is number, location, spherical

damping coefficients and transverse and axial moments of inertia of the liquid

assumed rigid. This data is input for the main simulation program via two input

files.

2. Output

Other quantities besides generalized speeds and generalized coordinates

are produced as output from the simulation program. The quantities which are

also useful in studying spacecraft motion include central angular momentum of

15

the system, several energy functions of the system, the velocity of the mass

center of the system and the nutation angle. The energy functions available are

the kinetic energy of the system, the energy dissipated through the viscous

spherical joints, and the work done by the motor and the external forces. All the

above quantities are produced as a function of time. It is the nutation angle as a

function of time that is used a stability parameter throughout this study.

The attitude motion derived from Chung's model though, is decoupled

frcm its orbital motion. Therefore, the above mentioned output parameters only

represent the motion due to the generalized active and inertia forces. Chung has

provided the equations to compute total motion, but these need to be solved for

each specific orbit if complete information regarding the spacecraft attitude is

desired.

D. SATELLITE CONFIGURATION

The satellite configuration used in the analysis is based on INTELSAT VI.

This satellite has four fuel and four oxidizer tanks located on the rotor,

symmetrically positioned around the axis of rotation. The placement of the rotor

and platform as well as tank location and dimensions are depicted in Figure 3.

The mass and dry inertia properties of the rotor and platform are listed in Table

I. These are the properties used in the unmodified simulation analysis. Fuel and

oxidizer properties are listed in Table 2.

16

PLATFORM

ROTOR q

YC

FTK02~~ =TO FT4K83 mTO

- - -- - - - - - - - - - - - - - - - - - - - - - - - - - - y -

Figue 3.FuelTan Arrngemnt or ITELST V

17R

TABLE 1. Mass and Inertia Properties of the Rotor and Platform

(Dry)

Platform Rotor

Mass 1058.8 kg 695.7 kg

Ixx 1587.1 kg m2 927.0 kg m2

1 _ _ 1518.3 kg m2 1166.2 kg m2

I_ _ _ 1529.4 kg m 2 973.7 kg m2

Ixy 0 0

Ixz -44.4 kg m2 6.1 kg m 2

Ivz 0 0

TABLE 2. Liquid Fuel Characteristics

Fuel Oxidizer

Viscosity 9.73 E-7 m2/s 2.92E-7 m2/s

Density 876.2 kg/m 3 1448.3 kg/m 3

18

III. LIQUID SLOSH MODELS

The first two sections of the chapter discusses the two models of liquid slosh,

the "rigid slug" model and Abramson's uniform gravity model. Section C

presents Slafer's postflight analysis of the LEASAT instability problem. Slafer's

comparison of the two models with actual flight data is also included.

A. RIGID SLUG MODEL

Chung's simulation models liquid propellant as two degree of freedom

pendulums attached to a rigid, spinning rotor, with one pendulum for each

propellant tank. This pendulum model (a "rigid slug") assumes that 100 percent

of the fluid to be a distributed spherical mass, pivoting about the center of the

tank. The pendulum mass is located at the fluid mass center. The location of this

mass center determines the length of the pendulum arm. The frequency of the

pendulum's oscillations is an input required by the simulation. Chung's model is

depicted in Figure 4.

Chung's model of the pendulum's motion is very dependent on the following

equation, which governs the angle q.

(IE + m * 12) 4 + CE (1 + (m * d * I * j22) q = 0

The variables in the above equation are either known or can be easily computed,

except for the the inertia of the pendulum, IE, which is a function of the natural

frequency. The undamped natural frequency of the endulum, cOn, is/ m* d* I

n= IE + m * 12

where Q is the rotor rotation speed. Solving for the inertia term results in

IE = m (0) 2 *d *1 - 121

The inertia term and hence the accuracy of the results is determined by the

user

19

given fundamental natural frequency. Chung's simulation provides a

recommended frequency based on the analysis of Abramson. This value is used

only to check the given value.

z

PROPELLANT TANK

ROTOR

Figure 4. Liquid Fuel Modelled as a Spherical Pendulum

B. UNIFORM GRAVITY MODEL

Abramson's model differs from the "rigid slug" model primarily in the fact

that only a portion of the fuel is modelled as a spherical pendulum, and the rest

as a point mass which is added to the dry properties of the rotor. Abramson's

analytical model is depicted in Figure 5 with a schematic and in Figure 6 with the

corresponding data curves. Figure 6 is taken from Reference 8. It is clear that

20

the fuel is separated into two different masses, one that is stationary and one that

is free to move as a pendulum. Entering the curves with fill fraction, the values

for pendulum mass and from that point mass, pendulum arm length, pendulum

hinge point location, point mass location and liquid fuel slosh frequency can be

extracted. All data extracted from the curves is normalized as a function of the

specific spacecraft parameters.

KEY:mfull - 4/3pita' 3

mo - mfull - mlml - pendulum masso - (g/a)AO.5a - radius of tank

Li - pendulum arm length ,

lo - location of pt. mass aI1 - location of hinge pt.h - height of liquid

ilLl

Iii4 mo

Figure 5. Abramson's Analytical Model

The fact that the pendulum's parameters vary as a function of fuel load

differentiates this model from the "rigid slug" model. By incorporating the data

21

from Abramson's curves into Chung's simulation, the simulation is improved.

The procedure how to do this is described in Chapter IV, Section C.

3.4

2.4

-3.0

2.0 1.0

.6 0.8 2.6

•- 1. W-- o0.8 ., - /. ..... ,,-1.-8

0.8\0.4- U

0.4 0.2- .. .. .... .. .. .... 1.2

0 .1 Q2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 6. Abramson's Curves for Analytical Model [Ref. 8]

In Slafer's postflight tests, the rigid slug model could not duplicate the

instability the LEASAT missions experienced. In fact, other shortcomings of the

"rigid slug" model were also noted at this time.

22

Using a uniform gravity model similar to Abramson's however, although not

predicting the exact results, Slafer did predict the mass and the frequency trends.

Figure 7 shows the normalized first mode frequencies. In making the analogy to

a mechanical pendulum, all modal frequencies of a pendulum should be

considered. Slafer does not present the higher modes however, as the pendulum

masses become very small for frequencies above the fundamental.

03 0.22

]= Rigid Slug

---- AbramsonU* 0.20 - - LEASAT Flight

U. 0.

0"

u. 0. 16

. 0.14Z 0 20 40 60 so 100

Fill Fraction

Figure 7. Normalized First Mode Frequency Comparison

The Abramson model predicted slightly higher first mode frequencies but

the general trend matches the actual flight results very closely. The "rigid slug"

model's accuracy for predicting first mode frequencies however, is in question.

Only for selected fill fractions is the "rigid slug" model close to the actual flight

23

data. The value for (o0 in this figure is (g/a)0.5 where g is the value for gravity

and a is the fuel tank radius. The values for g are 32.2 ft/sec 2 for the Abramson

uniform gravity model and d*Q22 for the rigid slug model and the LEASAT

flight data. Figure 8 shows the no.mnalized first mode masses.

1.2-

94E 1.0 -

I. -- Rigid SlugS 0.8--- Abramson

•--- LEASAT flight

0 0.6

U.

0.2-

E0= 0.0 . -

0 20 40 60 80 100

Fill Fraction, Percent

Figure 8. Normalized First Mode Mass Comparison

Throughout the entire range of fuel loading, the Abramson model's

normalized mass ratio is slightly larger, but matches the trend of the LEASAT

flight data very closely. The rigid slug model depicts that 100 percent of the fuel

is modelled as a pendulum.

24

IV. PROCEDURE

The procedures to generate the data for both the platform and the rotor

asymmetry analysis are described in this chapter. The procedural steps to

modify the simulation for the actual analysis are also described.

A. SYSTEMS PARAMETERS FOR ASYMMETRY ANALYSIS

The main simulation program uses dry inertia properties of the spacecraft

referenced to a specific point, and mass, positional and inertia data for each

individual fuel tank. Chung authored an additional program named CONVRT

that is used to calculate this data and convert it to a format usable by the main

simulation program. CONVRT places this data into an output file named

SYSPAR, which is used by INTLVI, the main simulation program. The primary

inputs required by CONVRT are the following

1. Rotor rotation speed

2. Natural frequency of fuel

3. Fill fraction of the tanks

4. Mass of the rotor and the platform

5. Location of the center of mass of the rotor and the platform

6. Moments of inertia of the rotor and the platform

The output of CONVRT is placed in SYSPAR, to be used as an input file for

INTLVI. The contents of SYSPAR include

1. Number and location of tanks

2. Mass and dry inertia properties of the rotor and platform

3. Mass and inertia properties of the fuel in each fuel tank

4. Spherical damping coefficients of the spherical pendulums

25

In addition, CONVRT places the following data into another output file,

named TYPE.

1. Wet spacecraft inertia properties

2. Natural slosh frequency of the fuel

3. Damping ratios

Although the data in TYPE is not used by the main simulation program, it is

used in this analysis. The wet spacecraft parameters for each of the fuel loads

are used to compute new dry properties for the platform and the rotor. These

wet spacecraft parameters are listed in Table 3.

TABLE 3. Wet Spacecraft Parameters for Fuel Modelled as

Spherical Pendulums

Fuel Load Fuel Load Fuel Load Fuel Load Fuel Load

15% 20% 26.2% 50% 75%

Mass 2183.1 2326.0 2503.1 3183.2 3897.5

Ixx 3950.6 4192.1 4469.5 5352.6 6029.5

IwY 3778.3 4105.6 4491.2 5786.6 6851.8

Izz 3939.6 4181.1 4458.5 5341.6 6018.5

Ixy 0.0 0.0 0.0 0.0 0.0

Ixz 38.3 38.3 38.3 38.3 38.3

lYZ 0.0 0.0 0.0 0.0 0.0

The actual rotor asymmetry analysis involves changing SYSPAR data prior

to running the main simulation program, substituting in data which gives the

desired inertia ratios. The specific data that is changed includes the dry

transverse moments of inertia of the rotor and the platform, and the axial

26

moment of inertia of the rotor. The values used are generated with a spreadsheet

program (SPRDSHT A) written for this purpose. SPRDSHT A generates new

dry platform and rotor data for four inertia ratios and five fuel loads. The

inertia ratios are 1.03, 1.05, 1.07 and 1.1. The five fuel loads to be examined

for each inertia ratio are 15, 20, 26.2, 50 and 75 percent of the total spacecraft

fuel capacity. These fuel loadings will be representative of the stability of a

generic satellite from apogee kick motor firing to end of life. The procedure

that SPRDSHT A uses to generate the dry inertia properties of the rotor and

platform is as follows

1. Compute the average wet system transverse moment of inertia for each fuel

load from the data obtained from output file TYPE. This data is listed in Table

3.

2. Multiply this average transverse moment of inertia by the desired inertia

ratio. This is the new wet axial moment of inertia of the rotor, Ir.

3. Because the main simulation uses dry properties, the amount the dry rotor

axial moment of inertia must be changed has to be computed. This is done by

subtracting the old wet axial moment of inertia of the rotor (Izz from Table 3

minus dry axial moment of inertia of the platform) from Ir. This quantity is Air.

4. The new dry axial moment of inertia of the rotor Idr, is computed by adding

old dry axial moment of inertia and Air.

5. So that the rule that the sum of two moments of inertia must be greater than

the third is not violated, the dry transverse moments of inertia of the rotor must

also be increased. To compute the amount of increase, the sum of the dry rotor

transverse moments of inertia is subtracted from the new dry rotor axial moment

of inertia, Idr. The result is divided by two and added to the old dry rotor

transverse moments of inertia. To achieve perfect symmetry, the smaller of the

two moments is increased to equal the larger.

27

6. An additional adjustment is made in order that the wet spacecraft moments of

inertia are not adversely effected by the previous step. The corresponding

platform transverse moments of inertia are decreased by the amount the rotor

transverse moments of inertia are increased.

7. These new dry platform and rotor moments of inertia are substituted into the

file SYSPAR to be used by the main simulation program.

B. SIMULATION PROCEDURES FOR ASYMMETRY ANALYSIS

The procedure for rotor asymmetry analysis involves changing file SYSPAR

data and running the main simulation program. After each run of the main

simulation, the nutation angle in the output file PLOT is examined to see if the

nutation angle reduces to zero or if it increases. If the nutation angle decreases

to zero, the system is considered passively stable. If the system is stable, rotor

asymmetry is increased and the procedure repeated until instability occurs. The

point at which this happens is the stability cutoff point. Once the stability cutoff

point is reached, procedure is repeated for each inertia ratio and fuel load.

The percent asymmetry is defined as the difference of the two dry rotor

transverse moments of inertia to the sum of the dry transverse moments of

inertia multiplied by 100. Percent asymmetry is performed in the simulation by

increasing 1xx of the dry rotor. The specific steps of the procedure follow. The

procedure is also depicted graphically in a flowchart in Figure 9.

1. Modify CONVRrs input file with fuel load, fundamental liquid slosh

frequency and rotation speed of the rotor. The rotor speed is assumed constant

throughout the study at 30 RPM.

2. Execute CONVRT.

28

3. Modify CONVRT's output file SYSPAR prior to running the main

simulation program with data from SPRDSHT A. Specifically, dry moments of

inertia are modified.

4. Execute the main simulation program INTLVI.

5. Check the output file PLOT to determine if nutation angle is decreasing to

zero. If the system appears to be marginally stable, increase the damping

coefficient exponents and run the simulation again. It was often necessary to

increase the damping coefficients by a factor of 1000 to determine definitively,

the nutation angle response over time.

6. If the system is stable, increment the dry rotor asymmetry and run the

simulation again. Once the stability cutoff point is located, change SYSPAR data

with data for the next inertia ratio

7. Go to Step 4 until inertia ratio is equal to 1.1.

8. After all the stability cutoff points for that fuel load are found, CONVRT's

input file is changed with the next fuel load and the entire procedure repeated.

The platform asymmetry analysis was done identically to the rotor

asymmetry analysis with few exceptions . The most important being that not

only was lxx of the dry platform increased , but Izz of the dry platform reduced,

while still satisfying the percent asymmetry equation. The motivation for this

deviation from the rotor asymmetry analysis was to allow a direct comparison of

the results with the work of Myers in Reference 7.

29

J Initial Condition

Fuel - IS%

%AymtyExecue CONVIInrmt

Inerterta RatotioO

yes SaerfStable?

75%?

Figure 9. Flowchart for Rotor Asymmetry Analysis Procedure

30

C. SIMULATION PROCEDURES FOR MODEL ENHANCEMENT

The modificaaon to Chung's simulation incorporates Abramson's model for

liquid slosh. Abramson's model changes the following input data for the main

simulation program.

1. Pendulum length

2. Center of mass of rotor location

3. Mass of dry rotor

4. Dry inertia properties

5. Slosh frequency

b. Liquid percentage

The first step is to use Abramson's curves to compute what percentage of the

liquid mass is to be modeled as spherical pendulums and what percentage is to be

added to the dry rotor. Another spreadsheet (SPRDSHT B) was written to

compute the amount of liquid that is sloshing as well as add the remaining fuel as

masses to the dry rotor using the parallel axis theorem. The fuel that was added

to the dry rotor was modelled as spheres equivalent in size to the mass of fuel

being added. SPRDSHT B also calculates liquid slosh frequency and pendulum

length using Abramson's curves.

The initial conditions of CONVRT are modified with the new dry rotor

mass, new location of the center of mass of the rotor, the pendulum arm length,

and dry moments of inertia from SPRDSHT B. CONVRT is then executed with

Abramson's recommended fuel slosh frequency and fuel percentage. The

objective of this part of the procedure is to obtain the wet spacecraft moments of

inertia. As mentioned previously, these are generated by CONVRT and can be

found in CONVRT's other output file, TYPE. The wet spacecraft moments of

inertia are then used in SPRDSHT A to obtain new dry spacecraft properties for

31

the various fuel loads and inertia ratios. These wet spacecraft properties for the

modified Chung program are found in Table 4.

TABLE 4. Wet Spacecraft Parameters for Fuel Modelled as

Spherical Pendulums (Modified Simulation)

Fuel Load Fuel Load Fuel Load Fuel Load Fuel Load

15% 20% 26.2% 50% 75%

Mass 2190.8 2334.1 2511.8 3194.0 3885.0

lxx 4006.8 4251.5 4539.9 5758.6 7268.4

Iyy 3878.5 4215.8 4638.6 5791.6 6851.2

lzz 3995.9 4241.8 4528.9 5742.7 7244.7

Ixy 0.0 0.0 0.0 0.0 0.0

lxz 38.3 38.3 38.3 38.3 38.3

0.0 0.0 0.0 0.0 0.0

As with the unmodified simulation, the output of SPRDSHT A is then used to

modify SYSPAR data prior to running the main simulation program. The

procedure to do this is identical as that for the unmodified model.

32

V. RESULTS AND ANALYSIS

This chapter will present plots of the rotor asymmetry analysis for both the

modified and unmodified simulations The boundary between stability and

instability will be plotted as a function of fuel load and rotor asymmetry. The

minimum inertia ratio for both rotor and platform asymmetry as a function of

fuel load will also be plotted,

A. SIMULATION RESULTS FOR ROTOR ASYMMETRv

The results for rotor asymmetry using Chung's "rigid slug" model are

depicted in Figure 10.

Stability Cutoff Curves16-14 - -

S 12 - - - Inert 1.0

E - .o

S '1.0

1.0

aR2 - " e - ,.- .40

0 10 20 30 40 50 60 70 80

% Fuel Load

Figure 10. Stability Cutoff Points for Inertia Ratios 1.03, 1.05, 1.07and 1.1.

The points are plotted midway between the last stable percent asymmetry and

the first unstable percent asymmetry. For an inertia ratio of 1.03, fuel loads of

15, 20 and 26.2 percent are unstable in their symmetric rotor configuration.

33

The results for rotor asymmetry using the modified version of Chung's

simulation are depicted in Figure It. Again, the points are plotted midway

between the last stable percent asymmetry and the first unstable percent

asymmetry.

Stability Cutoff Curves10-

• /i .Inertia RatiosE 7E. 6. . - m - - 1.1

5- 1.07I-4 - 1.05

o M--r- 1.03

01

0 10 20 30 40 50 60 70 80

% Fuel Load

Figure 11. Stability Cutoff Points for Inertia Ratios 1.03, 1.05, 1.07and 1.1. (Modified Simulation)

For an inertia ratio of 1.03, all fuel loads except 75 percent were unstable in

the symmetrical rotor configuration. For an inertia ratio of 1.05, the 20 percent

fuel load was unstable in its symmetric configuration. The line connecting the 15

and 26.2 percent fuel load points, shows only the trend and does not include any

intermediate points. The other two inertia ratios show a much more gradual

increase in allowable percent rotor asymmetry as fuel load increases, than the

unmodified Chung simulation.

34

Figure 12 depicts the minimum inertia ratio for zero, five and ten percent

asymmetry. As expected, significant increases in inertia ratio are required as the

rotor percent asymmetry is increased.

Minimum Inertia Ratio for Rotor Asymmetry

1.14

1.120S1.10 "

-----1.08.-- ---- Symmetric- --- 5% Rtr Asymt 1.06 - - 10% Rtr Asym0 __ _ _ _ _ __ _ _ _ _ _

- 1.0 4 -A 1-- "....

1.02 1

1.00. I 9

0 10 20 30 40 50 60 70 80 90 100% Fuel Load

Figure 12. Minimum Inertia Ratio for Stability (Rotor Asymmetry)

The results also indicate that the maximum inertia ratio required is between

20-40 percent fuel load. It was expected that the maximum inertia ratio would

be required at 50 percent fuel load because at this fuel load, liquid fuel surface

area is a maximum.

B. SIMULATION RESULTS FOR PLATFORM ASYMMETRY

The results for the platform asymmetry analysis are depicted in Figure 13.

The stability curve for zero percent asymmetry is identical to the one depicted in

the rotor asymmetry plot. The curves for five and ten percent platform

35

asymmetry however, show that for fuel loads of 50 and 70 percent, stability

increases as the percent platform asymmetry increases.

Minimum Inertia Ratio For Platform Asymmetry1.05-

O-.0 Symmetric

2 ---- 5% Pit Asymt 1.02 2- 10% Pit Asym

1.01 ,x

1.00

0 10 20 30 40 50 60 70 80 90 100% Fuel Load

Figure 13. Minimum Inertia Ratio for Stability (Platform

Asymmetry)

The five and ten percent asymmetry curves are generated by keeping the rotor

symmetric and varying the dry platform asymmetry. As with the minimum

inertia ratio for the rotor asymmetry plot, fuel loads of 10, 30, 40, 50, 70 and 90

were run.

C. ANALYSIS

For the rotor asymmetry analysis, both the modified and unmodified

simulations were run for five fuel loads and four inertia ratios. Data points are

plotted halfway between the last stable asymmetry percent and the first unstable

asymmetry percent. The data points that are not plotted were unstable in their

symmetric configuration.

36

The results of Chung's original "rigid slug" model depict relatively smooth,

increasing curves. The curves show that as the percentage of the fuel in the tanks

increases, so does the allowable percent rotor asymmetry. The curves also show

that as inertia ratio increases, the allowable percent asymmetry increases. This

result is in agreement with Likins' energy sink results, that is, inertia ratio must

be greater than one for passive stability. It is reasonable to assume that as inertia

ratio becomes greater than one, the spacecraft can tolerate a higher degree of

rotor asymmetry.

The curves of the modified simulation appear somewhat irregular and for

lower inertia ratios, broken. These results need to be assessed for each inertia

ratio. For inertia ratios of 1.1 and 1.07, the modified simulation comes into

very close agreement with Slafer's finding that propellant damping (with

unbaffled tanks) is very small. The curves are almost linear and is what would

be expected based on actual flight results. The curve is somewhat irregular

however, for an inertia ratio of 1.05. A more thorough discussion of the

modification procedures is required to explain this. First off, for low fuel loads,

Chung's and Abramson's models are nearly identical in terms of the mass size of

the pendulum. However, the length of the pendulum arms and the fundamental

frequencies are quite different. This difference can be seen in Table 5.

In substituting several Abramson frequencies and pendulum arm lengths,

CONVRT generated negative axial and transverse moments of inertia for the

spherical pendulums. This of course, is physically unrealizable. In order to

generate positive values for these inertias, it was decided to shorten the length of

the pendulum arms rather than to decrease the frequency of oscillation. If

CONVRT generated negative moments of inertias, the pendulum arm lengths

were reduce by five percent until positive moments were generated. The actual

pendulum arm lengths used in the study can be found in Table 6.

37

TABLE 5. Frequencies and Pendulum Arm Lengths of Models

Fundamental Length of

Frequency (rad/sec) Pendulum arm (meters)

Fuel Load (%) Chung Abramson Chung Abramson

15.0 6.4843 6.0619 .285 .351

20.0 6.4660 6.2236 .263 .336

26.2 6.4867 6.4771 .238 .322

50.0 6.7998 7.2175 .157 .282

75.0 7.9852 7.6959 .081 .218

TABLE 6. Recommended and Actual Pendulum Arm Lengths

Length of

Pendulum Arm (meters)

Fuel Load (%) Abramson Actually used Percent Change

15.0 .351 .351 0

20.0 .336 .319 5

26.2 .322 .306 5

50.0 .282 .240 15

75.0 .218 .218 0

38

The irregularity in the stability cutoff curves for an inertia ratios 1.05 can be

explained in the change required to Abramson's model for low fuel loads. For

an inertia ratio of 1.03, only a singular point was obtained at 75 percent fuel load

with no general trend. This discrepancy is related either to the fact that

frequency and arm lengths for Abramson's model produced physically

unrealizable results, or because 1.03 may be close to the actual minimum inertia

ratio required for stability.

The most obvious difference between the modified and unmodified

simulations though, is that the unmodified version often predicts stability when

the modified simulation does not. This is especially true at the lower fuel loads.

The unmodified simulation obviously assumes a higher dampening effect caused

by liquid slosh. The Abramson modified simulation appears more realistic based

on Slafer's finding that liquid slosh contributes less than 0.05% of the total

damping.

The platform asymmetry results show that as platform asymmetry increases, so

does stability for certain fuel loads. At fuel loads of 50 and 70 percent, the

minimum inertia ratio decreases for five and ten percent asymmetry by 0.01.

The platform asymmetry apparently acts as a damping mechanism for the

system. Also of note is the fact with an inertia ratio of 1.03, a symmetric

satellite with 50 percent fuel is unstable. Increasing the platform asymmetry to

ten percent however, makes the minimum inertia ratio less than 1.03. This

increase in platform asymmetry made an inherently unstable symmetric

spacecraft, stable.

39

VI. SUMMARY AND CONCLUSIONS

This chapter presents a synopsis of the results of both the modified and

unmodified computer simulations from Chapter V. Conclusions drawn from the

study are presented in Section B.

A. SUMMARY

The results of this analysis agree with Slafer in showing that the "rigid slug"

method of modelling liquid fuel is inadequate. Chung's "rigid slug" model

assumed the whole liquid is moving which is incorrect. Using this simulation in

its current form then, may result in determining incorrect stability boundaries

for various spacecraft configurations. The simulation can be improved using

Abramson's model however, resulting in improved prediction of attitude stability

conditions of dual-spin spacecraft.

B. CONCLUSIONS

Slafer's experience with LEASAT satellites is that the "rigid slug" model

gave incorrect predictions of attitude stability and that Abramson's model

followed closely with LEASAT flight data. The modified version of Chung's

simulation will give improved predictions of the attitude of asymmetric dual-spin

spacecraft with large liquid fractions. However, there is still a need to further

improve the liquid slosh model. Liquid slosh modelling should include

modelling the point masses as their actual shape, rather than symmetrical spheres

for more accurate dry rotor inertia calculations. This would yield more accurate

new dry rotor moments of inertia In addition, fuel slosh frequency and

pendulum arm length predictions need improvement. Using Abramson's

frequencies and pendulum arm lengths in Chung's simulation, resulted in

40

negative pendulum moments of inertia for some cases. Future work in this area

should determine the exact relationship between liquid slosh frequency and

pendulum arm length.

Future work should also more closely examine the effect of increasing the

damping coefficient exponents by 1000. The coefficients had to be increased to

determine definitively the trend of the nutation angle within 50 seconds of

simulation time. Factors less than 1000 required simulation times of 200 to 300

seconds to determine the trend, times which would necessitate an unreasonable

amount of CPU time for each simulation run.

Lastly, Likins' energy sink equation states that it does not matter in what

body the asymmetry is physically located, so long as the average transverse

moments of inertia do not change. The results here show otherwise. Increases

in platform asymmetry are stabilizing, while increases in rotor asymmetry are

destabilizing. The energy sink equation, as Chung states in Reference 2, should

be used with extreme caution for dual-spin stabilized spacecraft with large

fraction liquid loads

41

LIST OF REFERENCES

1. National Aeronautics and Space Administration, Special Project 106, TheDynamic Behavior of Liquids in Moving Containers - With Applications toSpace Vehicle Technology, by H. N. Abramson (ed.), 1966.

2. Stanford University, Final Report Project No. INTEL 465, Energy sinkAnalysis of Dual Spin Spacecraft, by Ting-Hong Chung, October 1985.

3. Thomson, W. T. and Reiter, G. S., "Motion of an Asymmetric Body withInternal Dissipation," AIAA Journal, v. 1, no. 6, pp. 1429-1430, June 1963.

4. Likins, Peter W., "Attitude Stability Criteria for Dual Spin Spacecraft, "Journal of Spacecraft and Rocket, v. 4, pp. 1638-1643, December 1967.

5. Spencer, T.M., "Energy-Sink Analysis for Asymmetric Dual SpinSpacecraft," Journal of Spacecraft and Rocket, v. 11, pp. 463-468, July1974.

6. Slafer, L.I., and Challoner, A.O., "Propellant Interaction with the PayloadControl System of Dual-Spin Spacecraft," Journal of Guidance and Control,v. 11, no. 4, pp. 343-351, July/August 1988.

7. Myers, J.W., The Effects of Liquid Propellant Motion on the AttitudeStability of Spin Stabilized Spacecraft, Master's Thesis, Naval PostgraduateSchool, Monterey California, March 1990.

8. Armstrong, G.L., and others, Handbook of Astronautical Engineering, 1sted., p. 14-20, McGraw-Hill Book Co., 1961.

42

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No. Copies

1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22314-6145

2. Library, Code 52 2Naval Postgraduate SchoolMonterey, California 93943-5100

3. Prof. B.N. Agrawal, Code AA/Ag 2Department of Aeronautics and AstronauticsMonterey, California 93943-5000

4. Prof. I.M. Ross, Code AA/Ro 1Department of Aeronautics and AstronauticsMonterey, California 93943-5000

5. LCDR Michael J. Szostak 2VP - 5MI 34099-5902

43